$f(x, y) = (\cos(3x - y), x\sin(y))$ $\text{curl}(f) = $
Solution: The formula for curl in two dimensions is $\text{curl}(f) = \dfrac{\partial Q}{\partial x} - \dfrac{\partial P}{\partial y}$, where $P$ is the $x$ -component of $f$ and $Q$ is the $y$ -component. Let's differentiate! $\begin{aligned} \dfrac{\partial Q}{\partial x} &= \dfrac{\partial}{\partial x} \left[ x\sin(y) \right] \\ \\ &= \sin(y) \\ \\ \dfrac{\partial P}{\partial y} &= \dfrac{\partial}{\partial y} \left[ \cos(3x - y) \right] \\ \\ &= \sin(3x - y) \end{aligned}$ Therefore: $\begin{aligned} \text{curl}(f) &= \sin(y) - \sin(3x - y) \end{aligned}$